Problem: $\dfrac{ -7t + 10u }{ -10 } = \dfrac{ 3t - 9v }{ 6 }$ Solve for $t$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -7t + 10u }{ -{10} } = \dfrac{ 3t - 9v }{ 6 }$ $-{10} \cdot \dfrac{ -7t + 10u }{ -{10} } = -{10} \cdot \dfrac{ 3t - 9v }{ 6 }$ $-7t + 10u = -{10} \cdot \dfrac { 3t - 9v }{ 6 }$ Multiply both sides by the right denominator. $-7t + 10u = -10 \cdot \dfrac{ 3t - 9v }{ {6} }$ ${6} \cdot \left( -7t + 10u \right) = {6} \cdot -10 \cdot \dfrac{ 3t - 9v }{ {6} }$ ${6} \cdot \left( -7t + 10u \right) = -10 \cdot \left( 3t - 9v \right)$ Distribute both sides ${6} \cdot \left( -7t + 10u \right) = -{10} \cdot \left( 3t - 9v \right)$ $-{42}t + {60}u = -{30}t + {90}v$ Combine $t$ terms on the left. $-{42t} + 60u = -{30t} + 90v$ $-{12t} + 60u = 90v$ Move the $u$ term to the right. $-12t + {60u} = 90v$ $-12t = 90v - {60u}$ Isolate $t$ by dividing both sides by its coefficient. $-{12}t = 90v - 60u$ $t = \dfrac{ 90v - 60u }{ -{12} }$ All of these terms are divisible by $6$ Divide by the common factor and swap signs so the denominator isn't negative. $t = \dfrac{ -{15}v + {10}u }{ {2} }$